Spin Chains and their Symmetries using the Bethe Ansatz.

Díaz Marrero, María (2025) Spin Chains and their Symmetries using the Bethe Ansatz. Bachelor's Thesis, Mathematics.

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Abstract

This thesis covers one of the earliest discovered quantum integrable models, the Heisenberg spin chain, which describes a one-dimensional quantum magnet. Taking advantage of the chain symmetries, the coordinate Bethe ansatz (CBA) is systematically applied, alongside the algebraic Bethe ansatz (ABA), which as its name indicates, uses an algebraic framework emphasizing the integrability structure of the model. Both approaches are employed to diagonalize the system, leading to the derivation of the Bethe equations, that are used to determine the energy spectrum. It is shown that both approaches yield identical Bethe equations. Furthermore, key results are presented, including an analysis of the thermodynamic limit and the explicit derivation of the ground state energy for the antiferromagnetic case. A generalization of the model is further explored, known as the XXZ chain, where anisotropy is introduced. Both coordinate and algebraic Bethe ansatz techniques are again applied to obtain the Bethe equations. Finally, the thesis discusses a relevant consequence arising from the integrability feature of both models, specifically from the XXZ model: the concept of quantum group. This new algebraic structure is introduced, and subsequently, the quantum group symmetry of the XXZ chain is identified, which provides a better understanding for the algebraic structures underlying quantum integrable models.

Item Type:	Thesis (Bachelor's Thesis)
Supervisor name:	Gorbe, T.F. and Veen, R.I. van der
Degree programme:	Mathematics
Thesis type:	Bachelor's Thesis
Language:	English
Date Deposited:	30 Apr 2025 06:55
Last Modified:	30 Apr 2025 06:55
URI:	https://fse.studenttheses.ub.rug.nl/id/eprint/35125

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